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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"5c3fc3862d916103b36d35dbd379e11ed7c15da1074abaaad4855e1425bdf28d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-629fc911ba71fa1d","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:         1;\n  1:         3,          12;\n  2:        15,         132,          162;\n  3:       105,        1464,         3807,          2592;\n  4:       945,       18216,        73791,        104544,         48600;\n  5:     10395,      257076,      1426491,       3117744,       2984850,        1049760;\n  6:    135135,     4090428,     28955961,      86178384,     126489600,       91014192,       25719120;\n  7:   2027025,    72707760,    627612138,    2373698304,    4725072225,     5190258384,     2987275788,     705438720;\n  8:  34459425,  1430626320,  14595625386,   67130989056,  169339914225,   251532888624,   220332128940,  105634409472,   21427701120;\n  9: 654729075, 30903517980, 364227431274, 1976078810592, 6049546971075, 11363060748924, 13430144106792, 9777133755648, 4016852157456, 714256704000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(24*n^3-36*n^2+12*n)*x-(81*n^3-162*n^2+81*n)*x^2)*P(n-2,x) + ((24*n^3-36*n^2+12*n)*x^2+(36*n^3-72*n^2+36*n)*x^3)*d/dx P(n-2,x) + (4*n^2-2*n-1+(12*n^2-12*n)*x)*P(n-1,x) - (12*n^2-12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (24*n^3 - 36*n^2 + 12*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*P[n - 2, x] + ((24*n^3 - 36*n^2 + 12*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 2*n - 1 + (12*n^2 - 12*n)*x)*P[n - 1, x] - (12*n^2 - 12*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (24*n^3 - 36*n^2 + 12*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*v[i - 2] + ((24*n^3 - 36*n^2 + 12*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 2*n - 1 + (12*n^2 - 12*n)*x)*v[i - 1] - (12*n^2 - 12*n)*x^2*deriv(v[i - 1], x))/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7fd16a32c438d522","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-07b12f78e33af42b","e-426adcccdbe1d7a9"],"text":"A059364 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A059364 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}